Abstract
In this paper we show several similarities among logic systems that deal simultaneously with deductive and quantitative inference. We claim it is appropriate to call the tasks those systems perform as Quantitative Logic Reasoning. Analogous properties hold throughout that class, for whose members there exists a set of linear algebraic techniques applicable in the study of satisfiability decision problems. In this presentation, we consider as Quantitative Logic Reasoning the tasks performed by propositional Probabilistic Logic; first-order logic with counting quantifiers over a fragment containing unary and limited binary predicates; and propositional Łukasiewicz Infinitely-valued Probabilistic Logic.
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Notes
- 1.
In computational logic tradition, variables are also called (syntactical) atoms, but to avoid confusion with the algebraic use of ‘atom’ as the smallest nonzero element of an algebra, we use here instead the term propositional symbol, or (atomic) proposition.
- 2.
Thus, valuations can be seen as homomorphisms of the set of formulas into the two element Boolean Algebra \(\{0,1\}\).
- 3.
While the presentation here stays on the syntactical level, in algebraic terms this notion can be seen as a probability measure over the free boolean algebra, in the sense of [30]. Recall that a measure on A is a function \(\tau : A \rightarrow [0, 1]\) which is additive for incompatibles and also satisfies \(\tau (1) = 1\). When A is finite, as in the case here, every \(a \in A\) equals the disjunction of the atoms it dominates, so \(\tau \) is uniquely determined by its value at the set of (algebraic) atoms of A. For every element \(a \in A\) the value of \(\tau (a)\) is the sum of the values \(\tau (e)\) for all atoms \(e \le a\).
- 4.
The notion of a “world”, can be understood via Stone duality, whereby homomorphisms of a boolean algebra A of events into the two element boolean algebra \(\{0,1\}\) are a dual counterpart of A, consisting of all possible evaluations of the events of A into \(\{0,1\}\), and can thus be identified with the set of possible worlds where these events take place.
- 5.
Note that the fragment mentioned here has the finite model property [41].
- 6.
First-order one- and two-variable fragments are decidable, but the coding of counting quantifiers employs several new variables, so decidability is not immediate; see Proposition 3.2.
- 7.
Available at http://cqu.sourceforge.net.
- 8.
- 9.
References
Andersen, K., and D. Pretolani. 2001. Easy cases of probabilistic satisfiability. Annals of Mathematics and Artificial Intelligence 33 (1): 69–91.
Baader, F., M. Buchheit, and B. Hollander. 1996. Cardinality restrictions on concepts. Artificial Intelligence 88 (1): 195–213.
Baader, F., S. Brandt, and C. Lutz. 2005. Pushing the EL envelope. In IJCAI05, 19th International Joint Conference on Artificial Intelligence, pp. 364–369.
Bertsimas, D., and J.N. Tsitsiklis. 1997. Introduction to Linear Optimization. Athena Scientific.
Biere, A. 2014. Lingeling essentials, a tutorial on design and implementation aspects of the the sat solver lingeling. In POS@ SAT, pp. 88. Citeseer.
Bona, G.D., and M. Finger. 2015. Measuring inconsistency in probabilistic logic: rationality postulates and dutch book interpretation. Artificial Intelligence 227: 140–164.
Bona, G.D., F.G. Cozman, and M. Finger. 2014. Towards classifying propositional probabilistic logics. Journal of Applied Logic 12(3):349–368. Special Issue on Combining Probability and Logic to Solve Philosophical Problems.
Boole, G. 1854. An Investigation on the Laws of Thought. London: Macmillan. Available on project Gutemberg at http://www.gutenberg.org/etext/15114.
Bova, S., and T. Flaminio. 2010. The coherence of Łukasiewicz assessments is NP-complete. International Journal of Approximate Reasoning 51 (3): 294–304.
Bulatov, A.A., and A. Hedayaty. 2015. Galois correspondence for counting quantifiers. Multiple-Valued Logic and Soft Computing 24 (5–6): 405–424.
Calvanese, D., G. De Giacomo, D. Lembo, M. Lenzerini, and R. Rosati (2005). DL-Lite: Tractable description logics for ontologies. In Proceedings of the 20th National Conference on Artificial Intelligence (AAAI 2005), vol. 5, pp. 602–607.
Cignoli, R., I. d’Ottaviano, and D. Mundici. 2000. Algebraic Foundations of Many-Valued Reasoning, Trends in Logic. Netherlands: Springer.
de Finetti, B. 1931. Sul significato soggettivo della probabilità. Fundamenta Mathematicae 17(1): 298–329. Translated into English as “On the Subjective Meaning of Probability”. In Probabilitàe Induzione, ed. P. Monari, and D. Cocchi, 291–321. Bologna: Clueb (1993).
de Finetti, B. 1937. La prévision: Ses lois logiques, ses sources subjectives. In Annales de l’institut Henri Poincaré, vol. 7:1, pp. 1–68. English translation by Henry E. Kyburg Jr., as “Foresight: Its Logical Laws, its Subjective Sources.” In H. E. Kyburg Jr., and H. E. Smokler, “Studies in Subjective Probability”, J. Wiley, New York, pp. 93–158, 1964. Second edition published by Krieger, New York, pp. 53–118, 1980.
de Finetti, B. 2017. Theory of probability: A critical introductory treatment. Translated by Antonio Machí and Adrian Smith. Wiley.
Eckhoff, J. 1993. Helly, Radon, and Carathéodory type theorems. In Handbook of Convex Geometry, Edited by P.M. Gruber, and J.M. Wills, pp. 389–448. Elsevier Science Publishers.
Eén, N. and N. Sörensson. 2003. An extensible SAT-solver. In SAT 2003, vol. 2919 LNCS, pp. 502–518. Springer.
Eén, N., and N. Sörensson. 2006. Translating pseudo-boolean constraints into sat. Journal on Satisfiability, Boolean Modeling and Computation 2 (1–4): 1–26.
Fagin, R., J.Y. Halpern, and N. Megiddo. 1990. A logic for reasoning about probabilities. Information and Computation 87: 78–128.
Finger, M., and G.D. Bona. 2011. Probabilistic satisfiability: Logic-based algorithms and phase transition. In Internatioinal Joint Congerence on Artificial Intelligence (IJCAI), Edited by T. Walsh, pp. 528–533. IJCAI/AAAI Press.
Finger, M., and G.D. Bona. 2017. Algorithms for deciding counting quantifiers over unary predicates. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, February 4–9, 2017, San Francisco, California, USA., Edited by S.P. Singh, and S. Markovitch, pp. 3878–3884. AAAI Press.
Finger, M., and G. De Bona. 2015. Probabilistic satisfiability: algorithms with the presence and absence of a phase transition. Annals of Mathematics and Artificial Intelligence 75 (3): 351–379.
Finger, M., and S. Preto. 2018. Probably half true: Probabilistic satisfiability over Łukasiewicz infinitely-valued logic. In preparation.
Georgakopoulos, G., D. Kavvadias, and C.H. Papadimitriou. 1988. Probabilistic satisfiability. Journal of Complexity 4 (1): 1–11.
Grädel, E., and M. Otto. 1999. On logics with two variables. Theoretical Computer Science 224 (1): 73–113.
Grädel, E., P.G. Kolaitis, and M.Y. Vardi. 1997. On the decision problem for two-variable first-order logic. The Bulletin of Symbolic Logic 3 (1): 53–69.
Hailperin, T. 1986. Boole’s Logic and Probability (Second enlarged edition ed.), vol. 85 Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland.
Hansen, P., and B. Jaumard. 2000. Probabilistic satisfiability. In Handbook of Defeasible Reasoning and Uncertainty Management Systems, Edited by J. Kohlas, and S. Moral, vol.5, pp. 321–367. Springer.
Hansen, P., B. Jaumard, G.-B.D. Nguetsé, and M.P. de Aragão. 1995. Models and algorithms for probabilistic and bayesian logic. In IJCAI, pp. 1862–1868.
Horn, A., and A. Tarski. 1948. Measures in boolean algebras. Transactions of the American Mathematical Society 64 (3): 467–497.
Jaumard, B., P. Hansen, and M.P. de Aragão. 1991. Column generation methods for probabilistic logic. INFORMS Journal on Computing 3 (2): 135–148.
Kavvadias, D., and C.H. Papadimitriou. 1990. A linear programming approach to reasoning about probabilities. Annals of Mathematics and Artificial Intelligence 1: 189–205.
Lindström, P. 1966. First order predicate logic with generalized quantifiers. Theoria 32 (3): 186–195.
Martin, B., F.R. Madelaine, and J. Stacho. 2015. Constraint satisfaction with counting quantifiers. SIAM Journal on Discrete Mathematics 29 (2): 1065–1113.
Mostowski, A. 1957. On a generalization of quantifiers. Fundamenta Mathematicae 44 (2): 12–36.
Mundici, D. 2006. Bookmaking over infinite-valued events. International Journal of Approximate Reasoning 43 (3): 223–240.
Mundici, D. 2011. Advanced Łukasiewicz calculus and MV-algebras, Trends in Logic. Netherlands: Springer.
Nilsson, N. 1986. Probabilistic logic. Artificial Intelligence 28 (1): 71–87.
Papadimitriou, C., and K. Steiglitz. 1998. Combinatorial Optimization: Algorithms and Complexity. Dover.
Pratt-Hartmann, I. 2005. Complexity of the two-variable fragment with counting quantifiers. Journal of Logic, Language and Information 14 (3): 369–395.
Pratt-Hartmann, I. 2008. On the computational complexity of the numerically definite syllogistic and related logics. The Bulletin of Symbolic Logic 14 (1): 1–28.
Schrijver, A. 1986. Theory of Linear and Integer Programming. New York: Wiley.
Walley, P., R. Pelessoni, and P. Vicig. 2004. Direct algorithms for checking consistency and making inferences from conditional probability assessments. Journal of Statistical Planning and Inference 126 (1): 119–151.
Warners, J.P. 1998. A linear-time transformation of linear inequalities into conjunctive normal form. Information Processing Letters 68 (2): 63–69.
Acknowledgements
We are very thankful to Daniele Mundici for several discussions on multi-valued logics; we would also like to thank two reviewers for their very detailed comments. This work was supported by Fapesp projects 2015/21880-4 and 2014/12236-1 and CNPq grant PQ 306582/2014-7.
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Finger, M. (2018). Quantitative Logic Reasoning. In: Carnielli, W., Malinowski, J. (eds) Contradictions, from Consistency to Inconsistency. Trends in Logic, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-319-98797-2_12
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